Start the single player Campaign mode under the Regular or higher difficulty level. Complete the "Ghost Stories", "Brave New World", "No Man's Land", and "Struck Down" missions to unlock the "Extinction Mode" option at the main menu.

Alternately, start Multiplayer Career mode and reach level 5.

Modern Warfare Reference
In the mission "Atlas Falls," you'll breach a recreational room. There, you should see an arcade machine called Call of Duty: Modern Warfare.